The Relationship between Risk and Rates of Return

In the preceding section, we saw that under the CAPM theory, beta is the appropriate measure of a stock's relevant risk. Now we must specify the relationship between risk and return: For a given level of risk as measured by beta, what rate of return should

FIGURE 3-11 Calculating a Beta Coefficient for Wal-Mart Stores

Historic Realized

Historic Realized

investors require to compensate them for bearing that risk? To begin, let us define the following terms:

r = expected rate of return on the ith stock. ri = required rate of return on the ith stock. Note that if r is less than ri, you would not purchase this stock, or you would sell it if you owned it. If r were greater than ri, you would want to buy the stock, because it looks like a bargain. You would be indifferent if r = ri. r = realized, after-the-fact return. One obviously does not know what r will be at the time he or she is considering the purchase of a stock. rRF = risk-free rate of return. In this context, rRF is generally measured by the return on long-term U.S. Treasury bonds. bi = beta coefficient of the ith stock. The beta of an average stock is bA = 1.0.

rM = required rate of return on a portfolio consisting of all stocks, which is called the market portfolio. rM is also the required rate of return on an average (bA = 1.0) stock.

RPm = (rM — rRF) = risk premium on "the market," and also on an average (b = 1.0) stock. This is the additional return over the risk-free rate required to compensate an average investor for assuming an average amount of risk. Average risk means a stock whose bi = bA = 1.0.

RPi = (rM — rRF)bi = (RPM)bi = risk premium on the ith stock. The stock's risk premium will be less than, equal to, or greater than the premium on an average stock, RPM, depending on whether its beta is less than, equal to, or greater than 1.0. If bi = bA = 1.0, then RPi = RPm.

The market risk premium, RPM, shows the premium investors require for bearing the risk of an average stock, and it depends on the degree of risk aversion that investors on average have.13 Let us assume that at the current time, Treasury bonds yield rRF = 6% and an average share of stock has a required return of rM = 11%. Therefore, the market risk premium is 5 percent:

It follows that if one stock were twice as risky as another, its risk premium would be twice as high, while if its risk were only half as much, its risk premium would be half as large. Further, we can measure a stock's relative riskiness by its beta coefficient. Therefore, the risk premium for the ith stock is:

If we know the market risk premium, RPM, and the stock's risk as measured by its beta coefficient, bi, we can find the stock's risk premium as the product (RPM)bi. For example, if bi = 0.5 and RPM = 5%, then RPi is 2.5 percent:

As the discussion in Chapter 1 implied, the required return for any investment can be expressed in general terms as

Required return = Risk-free return + Premium for risk.

Here the risk-free return includes a premium for expected inflation, and we assume that the assets under consideration have similar maturities and liquidity. Under these conditions, the relationship between the required return and risk is called the Security Market Line (SML)

13It should be noted that the risk premium of an average stock, rM — rRF, cannot be measured with great precision because it is impossible to obtain precise values for the expected future return on the market, rM. However, empirical studies suggest that where long-term U.S. Treasury bonds are used to measure rRF and where rM is an estimate of the expected (not historical) return on the S&P 500 Industrial Stocks, the market risk premium varies somewhat from year to year, and it has generally ranged from 4 to 6 percent during the last 20 years.

If some other Stock j were riskier than Stock i and had bj = 2.0, then its required rate of return would be 16 percent:

An average stock, with b = 1.0, would have a required return of 11 percent, the same as the market return:

As noted above, Equation 3-9 is called the Security Market Line (SML) equation, and it is often expressed in graph form, as in Figure 3-12, which shows the SML when rRF = 6% and rM = 11%. Note the following points:

1. Required rates of return are shown on the vertical axis, while risk as measured by beta is shown on the horizontal axis. This graph is quite different from the one shown in Figure 3-9, where the returns on individual stocks were plotted on the vertical axis and returns on the market index were shown on the horizontal axis. The slopes of the three lines in Figure 3-9 were used to calculate the three stocks' betas, and those betas were then plotted as points on the horizontal axis of Figure 3-12.

2. Riskless securities have bi = 0; therefore, rRF appears as the vertical axis intercept in Figure 3-12. If we could construct a portfolio that had a beta of zero, it would have an expected return equal to the risk-free rate.

3. The slope of the SML (5% in Figure 3-12) reflects the degree of risk aversion in the economy—the greater the average investor's aversion to risk, then (a) the steeper the slope of the line, (b) the greater the risk premium for all stocks, and (c) the higher the required rate of return on all stocks.14 These points are discussed further in a later section.

4. The values we worked out for stocks with bi = 0.5, bi = 1.0, and bi = 2.0 agree with the values shown on the graph for rL, rA, and rH.

Both the Security Market Line and a company's position on it change over time due to changes in interest rates, investors' aversion to risk, and individual companies' betas. Such changes are discussed in the following sections.

14Students sometimes confuse beta with the slope of the SML. This is a mistake. The slope of any straight line is equal to the "rise" divided by the "run," or (Yj — Y0)/(Xj — X0). Consider Figure 3-12. If we let Y = r and X = beta, and we go from the origin to b = 1.0, we see that the slope is (rM — rRp)/(bM — bRp) = (11% — 6%)/(1 — 0) = 5%. Thus, the slope of the SML is equal to (rM — r^), the market risk premium. In Figure 3-12, ri = 6% + 5%bi, so an increase of beta from 1.0 to 2.0 would produce a 5 percentage point increase in ri.

FIGURE 3-13 Shift in the SML Caused by an Increase in Inflation Required Rate of Return (%)

FIGURE 3-13 Shift in the SML Caused by an Increase in Inflation Required Rate of Return (%)